Herd Immunity Threshold

When reading an article about vaccines and vaccine preventable diseases it's not uncommon for there to be a statement in the article along the lines of, "With 90% of full immunisation needed for herd immunity..." or, "When the number of people vaccinated drops below 95%, a community loses herd immunity to highly contagious germs like pertussis..." What are these numbers and how are they determined?

The numbers are referring to the 'herd immunity threshold' which is the approximate proportion of the population that's needed to be immunised so that the infectious disease will die out  in that population and it is worked out with the following formula:

threshold = 1 - 1/R₀

The key to understanding this formula is the 'basic reproduction number' (R₀). The basic reproduction number is the average number of additional people that will be infected for each case in a population with no immunity.

Figure 1: A disease with an R₀ of 4

If a disease has a reproduction number greater than 1 then the disease will increase in numbers in the population whereas if the reproduction number is less than 1 then eventually the disease will die out in the population. To create a population in which a disease cannot persist then it is necessary to reduce the reproduction number to less than 1 (or thereabouts).

Figure 2: A disease with an effective reproductive number reduced to 1

So out of all the number of additional people that would be infected from a case (R₀), all of them except one needs to be immunised (R₀-1), or as a proportion this can be written as:

(R₀-1)/R₀

Which can be arranged to arrive at the herd immunity threshold formula:

=> R₀/R₀ - 1/R₀

=> 1 - 1/R₀

The final figure given is an approximate figure and doesn't take into account things like vaccine effectiveness or other public health efforts used to combat infectious diseases. Finally, herd immunity is not all or nothing and even if the vaccination rates aren't as high as the herd immunity threshold it's still possible for unvaccinated people to benefit from the cocooning effect of herd immunity. If you vaccinate you're not only protecting yourself, you're also helping to protect those who can't (or won't) be vaccinated.